CW1_12711142_A_RUSSELL
CW1_12711142_A_RUSSELL
06 January 2019
BDA/IDAR Course Work 2
Adam Russell, 12711142, MSc Data Science (Part time)
1. Decision Trees
(a)
(b)
2. Regression Trees
library(MASS)## Warning: package 'MASS' was built under R version 3.5.2library(ISLR)
library(tree)## Warning: package 'tree' was built under R version 3.5.2fix(Carseats)
?Carseats## starting httpd help server ... doneset.seed(1)
#(a)
Carseats_split <- sample(1:nrow(Carseats), nrow(Carseats)/2)
Carseats.train <- Carseats[Carseats_split,]
Carseats.test <- Carseats[-Carseats_split,]
#(b)
tree.Carseats <- tree(Carseats$Sales ~ .,data = Carseats, subset = Carseats_split)
summary(tree.Carseats)##
## Regression tree:
## tree(formula = Carseats$Sales ~ ., data = Carseats, subset = Carseats_split)
## Variables actually used in tree construction:
## [1] "ShelveLoc" "Price" "Age" "Advertising" "Income"
## [6] "CompPrice"
## Number of terminal nodes: 18
## Residual mean deviance: 2.36 = 429.5 / 182
## Distribution of residuals:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -4.2570 -1.0360 0.1024 0.0000 0.9301 3.9130#The tree function has only used: shelve location, price, average age of location population, local advertising budget
#community income and competitor price at each location to model indicating that education level, urban or rural, US or not
#and population size in region are not significant in relation to unit sales of car seats.
plot(tree.Carseats)
text(tree.Carseats,pretty = 0)
#The plot of the regression decision tree indicates a good shelve location to be the biggest driver of unit sales of carseats.
#The second critical driver is Carseat price.
yhat <- predict(tree.Carseats, newdata = Carseats[-Carseats_split,])
Carseats.test_sales <- Carseats[-Carseats_split, "Sales"]
mean((yhat-Carseats.test_sales)^2) #test MSE## [1] 4.148897#(c)
cv.Carseats <- cv.tree(tree.Carseats)
plot(cv.Carseats$size, cv.Carseats$dev,type = 'b')
#The CV plot indicates that the minimum MSE is when the number of lead nodes is 7.
prune.Carseats <- prune.tree(tree.Carseats,best=7)
plot(prune.Carseats)
text(prune.Carseats,pretty = 0)
yhat_prune <- predict(prune.Carseats, newdata = Carseats[-Carseats_split,])
mean((yhat_prune-Carseats.test_sales)^2) #test MSE## [1] 5.340397#there for prunning the tree to 7 terminal nodes from 18 did not reduce the test MSE indicating the model
#has high bias.
#(d)
library(randomForest)## Warning: package 'randomForest' was built under R version 3.5.2## randomForest 4.6-14
## Type rfNews() to see new features/changes/bug fixes.bag.Carseats = randomForest(Carseats$Sales ~ .,data = Carseats, subset = Carseats_split, mtry = 10, importance = TRUE)
bag.Carseats##
## Call:
## randomForest(formula = Carseats$Sales ~ ., data = Carseats, mtry = 10, importance = TRUE, subset = Carseats_split)
## Type of random forest: regression
## Number of trees: 500
## No. of variables tried at each split: 10
##
## Mean of squared residuals: 2.811039
## % Var explained: 63.17yhat_bag <- predict(bag.Carseats, newdata = Carseats[-Carseats_split,])
mean((yhat_bag-Carseats.test_sales)^2) #test MSE## [1] 2.633915#Large reduction in test MSE to 2.633915 when compared with pruned tree
importance(bag.Carseats)## %IncMSE IncNodePurity
## CompPrice 16.9874366 126.852848
## Income 3.8985402 78.314126
## Advertising 16.5698586 123.702901
## Population 0.6487058 62.328851
## Price 55.3976775 514.654890
## ShelveLoc 42.7849818 319.133777
## Age 20.5135255 185.582077
## Education 3.4615211 42.253410
## Urban -2.5125087 8.700009
## US 7.3586645 18.180651varImpPlot(bag.Carseats)
#We can see that the biggest drop in prediction accuracy is when the variables Price and Shelve Location are excluded
#which was observed in (b)
#(e)
testMSE <- rep(0,10)
for (i in 1:10) {
set.seed(1)
rf.Carseats = randomForest(Carseats$Sales ~ .,data = Carseats, subset = Carseats_split, mtry = i, importance = TRUE)
yhat_rf <- predict(rf.Carseats, newdata = Carseats[-Carseats_split,])
testMSE[i] <- mean((yhat_rf-Carseats.test_sales)^2) #test MSE
}
plot(testMSE, type = "b", xlab = "mtry", ylab = "Test MSE")
#The above plot demonstrates that the random forest model is no better then the bagged model suggesting
#the variables of the careseat data set are not highly correlated.
rf.Carseats = randomForest(Carseats$Sales ~ .,data = Carseats, subset = Carseats_split, mtry = 8, importance = TRUE)
importance(rf.Carseats)## %IncMSE IncNodePurity
## CompPrice 13.95307865 130.822017
## Income 5.01601324 80.900674
## Advertising 12.80534252 128.915509
## Population -0.09912927 66.842641
## Price 54.75918061 496.543261
## ShelveLoc 42.94342059 309.717773
## Age 20.95271103 194.037615
## Education 2.27870853 42.731861
## Urban -2.07631988 9.806819
## US 6.41852889 18.554327varImpPlot(rf.Carseats)
#And accordingly the variable importance conclusions for the random forest model are the same as for the bagged
#model in part (d).3. Classification Trees
library(MASS)
library(ISLR)
library(tree)
library(randomForest)
fix(OJ)
?OJ
set.seed(1)
#(a)
OJ_split <- sample(1:nrow(OJ), 800)
OJ.train <- OJ[OJ_split,]
OJ.test <- OJ[-OJ_split,]
#(b)
tree.OJ <- tree(Purchase ~., data=OJ, subset = OJ_split)
summary(tree.OJ)##
## Classification tree:
## tree(formula = Purchase ~ ., data = OJ, subset = OJ_split)
## Variables actually used in tree construction:
## [1] "LoyalCH" "PriceDiff" "SpecialCH" "ListPriceDiff"
## Number of terminal nodes: 8
## Residual mean deviance: 0.7305 = 578.6 / 792
## Misclassification error rate: 0.165 = 132 / 800#The summary indicates that only customer brand loyalty, sale price difference between MM and CH, Sale price for CH and List price difference
#are signifcant predictors In CH or MM purchase. There are 8 terminal nodes. The training error rate is 0.165 = 132 / 800.
#(c)
tree.OJ## node), split, n, deviance, yval, (yprob)
## * denotes terminal node
##
## 1) root 800 1064.00 CH ( 0.61750 0.38250 )
## 2) LoyalCH < 0.508643 350 409.30 MM ( 0.27143 0.72857 )
## 4) LoyalCH < 0.264232 166 122.10 MM ( 0.12048 0.87952 )
## 8) LoyalCH < 0.0356415 57 10.07 MM ( 0.01754 0.98246 ) *
## 9) LoyalCH > 0.0356415 109 100.90 MM ( 0.17431 0.82569 ) *
## 5) LoyalCH > 0.264232 184 248.80 MM ( 0.40761 0.59239 )
## 10) PriceDiff < 0.195 83 91.66 MM ( 0.24096 0.75904 )
## 20) SpecialCH < 0.5 70 60.89 MM ( 0.15714 0.84286 ) *
## 21) SpecialCH > 0.5 13 16.05 CH ( 0.69231 0.30769 ) *
## 11) PriceDiff > 0.195 101 139.20 CH ( 0.54455 0.45545 ) *
## 3) LoyalCH > 0.508643 450 318.10 CH ( 0.88667 0.11333 )
## 6) LoyalCH < 0.764572 172 188.90 CH ( 0.76163 0.23837 )
## 12) ListPriceDiff < 0.235 70 95.61 CH ( 0.57143 0.42857 ) *
## 13) ListPriceDiff > 0.235 102 69.76 CH ( 0.89216 0.10784 ) *
## 7) LoyalCH > 0.764572 278 86.14 CH ( 0.96403 0.03597 ) *#Node 21 is a terminal node:
# 21) SpecialCH > 0.5 13 16.05 CH ( 0.69231 0.30769 ) *
#All the observations on this branch have a 'Indicator of special on CH' greater then 0.5.
#There are 13 oberservations on this branch.
#deviance = 16.05
#The overall prediction for this branch is a purchase of CH as indicated by 0.69231 of the observations at this node.
#(d)
plot(tree.OJ)
text(tree.OJ,pretty = 0)
#The resulting classification decision tree indicates that customers with a CH brand loyalty greater then 0.508643 are highly likely to purchase
#CH over MM. This is indicated by ALL the terminal nodes following LoyalCH < 0.508643 split along the NO branch predicting CH purchases. Conversely customers
#with CH brand loyalty less then 0.264232 are highley likely to purchase MM.This is indicated by ALL the terminal nodes following
#LoyalCH < 0.264232 split along the YES branch predicting MM purchases.Customers with LoyalCH between 0.26232 and 0.508643 make a decision between MM and CH
#based on price difference and if there is a special deal on CH.
#(e)
tree.OJ.pred <- predict(tree.OJ, OJ.test, type = "class")
OJ.test_Purchase <- OJ[-OJ_split, "Purchase"]
table(tree.OJ.pred, OJ.test_Purchase)## OJ.test_Purchase
## tree.OJ.pred CH MM
## CH 147 49
## MM 12 62(147+62)/270## [1] 0.7740741#The test error rate is 0.7740741
#(f)
cv.OJ <- cv.tree(tree.OJ, FUN=prune.misclass)
names(cv.OJ)## [1] "size" "dev" "k" "method"cv.OJ## $size
## [1] 8 5 2 1
##
## $dev
## [1] 156 156 156 306
##
## $k
## [1] -Inf 0.000000 4.666667 160.000000
##
## $method
## [1] "misclass"
##
## attr(,"class")
## [1] "prune" "tree.sequence"#(g)
par(mfrow=c(1,2))
plot(cv.OJ$size,cv.OJ$dev,type="b")
#(h)
#The results indicate that a classitication decision tree with two or more terminal nodes have lowest error rate.
#(i)
#Tree pruning to 5 terminal nodes was selected as the cross validation classification error is inidcated as being the same
#for 2 or more terminal nodes.
prune.OJ <- prune.misclass(tree.OJ, best = 5)
plot(prune.OJ)
text(prune.OJ,pretty=0)
#(j)
summary(prune.OJ)##
## Classification tree:
## snip.tree(tree = tree.OJ, nodes = 3:4)
## Variables actually used in tree construction:
## [1] "LoyalCH" "PriceDiff" "SpecialCH"
## Number of terminal nodes: 5
## Residual mean deviance: 0.8256 = 656.4 / 795
## Misclassification error rate: 0.165 = 132 / 800#The training error rate is the same as for the unpruned tree: 0.165 = 132/800
#(k)
prune.OJ.pred <- predict(prune.OJ, OJ.test, type = "class")
table(prune.OJ.pred, OJ.test_Purchase)## OJ.test_Purchase
## prune.OJ.pred CH MM
## CH 147 49
## MM 12 62(147+62)/270## [1] 0.7740741#The test error rate is the same as for the unpruned tree: 0.77407414.SVM
library(MASS)
library(ISLR)
library(e1071)## Warning: package 'e1071' was built under R version 3.5.2library(LiblineaR)## Warning: package 'LiblineaR' was built under R version 3.5.2set.seed(1)
fix(Auto)
?Auto
#(a)
above_median_mpg <- ifelse(Auto$mpg<=median(Auto$mpg),0,1)
median(Auto$mpg)## [1] 22.75Auto2 <- data.frame(xx=Auto[0:6], yy=as.factor(above_median_mpg))
#Removing name, year of manufacture, cylinders and origin information from data Auto dataset.
head(Auto2)## xx.mpg xx.cylinders xx.displacement xx.horsepower xx.weight
## 1 18 8 307 130 3504
## 2 15 8 350 165 3693
## 3 18 8 318 150 3436
## 4 16 8 304 150 3433
## 5 17 8 302 140 3449
## 6 15 8 429 198 4341
## xx.acceleration yy
## 1 12.0 0
## 2 11.5 0
## 3 11.0 0
## 4 12.0 0
## 5 10.5 0
## 6 10.0 0#(b)
Auto2.svm.tune.linear <- tune(svm, yy~., data=Auto2, kernel="linear", ranges=list(cost=c(0.001 , 0.01 , 0.1, 1 ,5 ,10 ,100, 1000)))
summary(Auto2.svm.tune.linear)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost
## 5
##
## - best performance: 0.0025
##
## - Detailed performance results:
## cost error dispersion
## 1 1e-03 0.122692308 0.054156614
## 2 1e-02 0.081730769 0.055230186
## 3 1e-01 0.053589744 0.045888164
## 4 1e+00 0.017820513 0.024132627
## 5 5e+00 0.002500000 0.007905694
## 6 1e+01 0.005064103 0.010677135
## 7 1e+02 0.005064103 0.010677135
## 8 1e+03 0.005064103 0.010677135#The results of the tuning show that the errors are at a minium for a cost equal to 5.
Auto2.svm.linear <- svm(formula = yy~., data = Auto2, kernel = "linear", cost = 5)
summary(Auto2.svm.linear)##
## Call:
## svm(formula = yy ~ ., data = Auto2, kernel = "linear", cost = 5)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: linear
## cost: 5
## gamma: 0.1666667
##
## Number of Support Vectors: 25
##
## ( 12 13 )
##
##
## Number of Classes: 2
##
## Levels:
## 0 1#(c)
Auto2.svm.tune.radial <- tune(svm, yy~., data=Auto2, kernel="radial", ranges=list(cost=c(0.001 , 0.01 , 0.1, 1 ,5 ,10 ,100, 1000),gamma =c(0.001 , 0.01 , 0.1, 1 ,5 ,10 ,100, 1000)))
summary(Auto2.svm.tune.radial)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost gamma
## 1000 0.001
##
## - best performance: 0.002564103
##
## - Detailed performance results:
## cost gamma error dispersion
## 1 1e-03 1e-03 0.558717949 0.050683112
## 2 1e-02 1e-03 0.558717949 0.050683112
## 3 1e-01 1e-03 0.558717949 0.050683112
## 4 1e+00 1e-03 0.097051282 0.061308926
## 5 5e+00 1e-03 0.086794872 0.041988348
## 6 1e+01 1e-03 0.079102564 0.032595382
## 7 1e+02 1e-03 0.030641026 0.010881923
## 8 1e+03 1e-03 0.002564103 0.008108404
## 9 1e-03 1e-02 0.558717949 0.050683112
## 10 1e-02 1e-02 0.558717949 0.050683112
## 11 1e-01 1e-02 0.094487179 0.060390528
## 12 1e+00 1e-02 0.081666667 0.035523079
## 13 5e+00 1e-02 0.056025641 0.035213260
## 14 1e+01 1e-02 0.028012821 0.014346664
## 15 1e+02 1e-02 0.007692308 0.012385792
## 16 1e+03 1e-02 0.007692308 0.012385792
## 17 1e-03 1e-01 0.558717949 0.050683112
## 18 1e-02 1e-01 0.091923077 0.061748218
## 19 1e-01 1e-01 0.081666667 0.035523079
## 20 1e+00 1e-01 0.048461538 0.025333267
## 21 5e+00 1e-01 0.010256410 0.013240969
## 22 1e+01 1e-01 0.007692308 0.012385792
## 23 1e+02 1e-01 0.012756410 0.013447795
## 24 1e+03 1e-01 0.007628205 0.012283816
## 25 1e-03 1e+00 0.558717949 0.050683112
## 26 1e-02 1e+00 0.558717949 0.050683112
## 27 1e-01 1e+00 0.084230769 0.036063920
## 28 1e+00 1e+00 0.030576923 0.033470093
## 29 5e+00 1e+00 0.022884615 0.021941126
## 30 1e+01 1e+00 0.022948718 0.022275167
## 31 1e+02 1e+00 0.030576923 0.023148137
## 32 1e+03 1e+00 0.030576923 0.023148137
## 33 1e-03 5e+00 0.558717949 0.050683112
## 34 1e-02 5e+00 0.558717949 0.050683112
## 35 1e-01 5e+00 0.161025641 0.075794667
## 36 1e+00 5e+00 0.061217949 0.029803288
## 37 5e+00 5e+00 0.066410256 0.030127750
## 38 1e+01 5e+00 0.066410256 0.030127750
## 39 1e+02 5e+00 0.066410256 0.030127750
## 40 1e+03 5e+00 0.066410256 0.030127750
## 41 1e-03 1e+01 0.558717949 0.050683112
## 42 1e-02 1e+01 0.558717949 0.050683112
## 43 1e-01 1e+01 0.556153846 0.053814941
## 44 1e+00 1e+01 0.094551282 0.034593285
## 45 5e+00 1e+01 0.097179487 0.041840204
## 46 1e+01 1e+01 0.097179487 0.041840204
## 47 1e+02 1e+01 0.097179487 0.041840204
## 48 1e+03 1e+01 0.097179487 0.041840204
## 49 1e-03 1e+02 0.558717949 0.050683112
## 50 1e-02 1e+02 0.558717949 0.050683112
## 51 1e-01 1e+02 0.558717949 0.050683112
## 52 1e+00 1e+02 0.466602564 0.095169918
## 53 5e+00 1e+02 0.448717949 0.090453036
## 54 1e+01 1e+02 0.448717949 0.090453036
## 55 1e+02 1e+02 0.448717949 0.090453036
## 56 1e+03 1e+02 0.448717949 0.090453036
## 57 1e-03 1e+03 0.558717949 0.050683112
## 58 1e-02 1e+03 0.558717949 0.050683112
## 59 1e-01 1e+03 0.558717949 0.050683112
## 60 1e+00 1e+03 0.535641026 0.069585458
## 61 5e+00 1e+03 0.533076923 0.072000114
## 62 1e+01 1e+03 0.533076923 0.072000114
## 63 1e+02 1e+03 0.533076923 0.072000114
## 64 1e+03 1e+03 0.533076923 0.072000114#The results of the tuning show that the errors are at a minium for costs equal to 1000 and gamma equal to 0.001.
Auto2.svm.radial <- svm(formula = yy~., data = Auto2, gamma = 0.001, kernel = "radial", cost = 1000)
summary(Auto2.svm.radial)##
## Call:
## svm(formula = yy ~ ., data = Auto2, gamma = 0.001, kernel = "radial",
## cost = 1000)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: radial
## cost: 1000
## gamma: 0.001
##
## Number of Support Vectors: 36
##
## ( 17 19 )
##
##
## Number of Classes: 2
##
## Levels:
## 0 1Auto2.svm.tune.poly <- tune(svm, yy~., data=Auto2, kernel="polynomial", ranges=list(cost=c(0.001 , 0.01 , 0.1, 1 ,5 ,10 ,100, 1000),degree=c(2,3,4,5)))
summary(Auto2.svm.tune.poly)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost degree
## 1000 3
##
## - best performance: 0.02544872
##
## - Detailed performance results:
## cost degree error dispersion
## 1 1e-03 2 0.58673077 0.03560377
## 2 1e-02 2 0.45948718 0.09665151
## 3 1e-01 2 0.29314103 0.06439963
## 4 1e+00 2 0.24730769 0.05790701
## 5 5e+00 2 0.24724359 0.07300194
## 6 1e+01 2 0.25237179 0.07077783
## 7 1e+02 2 0.22961538 0.06285452
## 8 1e+03 2 0.19634615 0.06809105
## 9 1e-03 3 0.35211538 0.11377480
## 10 1e-02 3 0.25506410 0.05294702
## 11 1e-01 3 0.22192308 0.06855266
## 12 1e+00 3 0.08416667 0.02420111
## 13 5e+00 3 0.06128205 0.02159748
## 14 1e+01 3 0.04339744 0.02403832
## 15 1e+02 3 0.02807692 0.02537217
## 16 1e+03 3 0.02544872 0.01688453
## 17 1e-03 4 0.43358974 0.10882896
## 18 1e-02 4 0.34942308 0.07545637
## 19 1e-01 4 0.26262821 0.05515474
## 20 1e+00 4 0.25224359 0.06429973
## 21 5e+00 4 0.24455128 0.06928960
## 22 1e+01 4 0.22423077 0.06216796
## 23 1e+02 4 0.25224359 0.04818388
## 24 1e+03 4 0.20141026 0.05538501
## 25 1e-03 5 0.36730769 0.08638189
## 26 1e-02 5 0.28057692 0.06415813
## 27 1e-01 5 0.24993590 0.05252452
## 28 1e+00 5 0.19871795 0.06975702
## 29 5e+00 5 0.13006410 0.02518913
## 30 1e+01 5 0.11730769 0.02733039
## 31 1e+02 5 0.08929487 0.03230957
## 32 1e+03 5 0.05096154 0.03751354#The results of the tuning show that the errors are at a minium for costs equal to 1000 and degree equal to 3.
Auto2.svm.poly <- svm(formula = yy~., data = Auto2, kernel = "polynomial", cost = 1000, degree = 3)
summary(Auto2.svm.poly)##
## Call:
## svm(formula = yy ~ ., data = Auto2, kernel = "polynomial", cost = 1000,
## degree = 3)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: polynomial
## cost: 1000
## degree: 3
## gamma: 0.1666667
## coef.0: 0
##
## Number of Support Vectors: 35
##
## ( 17 18 )
##
##
## Number of Classes: 2
##
## Levels:
## 0 1#(d)
plot(Auto2.svm.linear, Auto2, xx.mpg ~ xx.cylinders)
plot(Auto2.svm.radial, Auto2, xx.mpg ~ xx.cylinders)
plot(Auto2.svm.poly, Auto2, xx.mpg~ xx.cylinders)
plot(Auto2.svm.linear, Auto2, xx.mpg ~ xx.displacement)
plot(Auto2.svm.radial, Auto2, xx.mpg ~ xx.displacement)
plot(Auto2.svm.poly, Auto2, xx.mpg~ xx.displacement)
plot(Auto2.svm.linear, Auto2, xx.mpg ~ xx.horsepower)
plot(Auto2.svm.radial, Auto2, xx.mpg ~ xx.horsepower)
plot(Auto2.svm.poly, Auto2, xx.mpg~ xx.horsepower)
plot(Auto2.svm.linear, Auto2, xx.mpg ~ xx.weight)
plot(Auto2.svm.radial, Auto2, xx.mpg ~ xx.weight)
plot(Auto2.svm.poly, Auto2, xx.mpg~ xx.weight)
plot(Auto2.svm.linear, Auto2, xx.mpg ~ xx.acceleration)
plot(Auto2.svm.radial, Auto2, xx.mpg ~ xx.acceleration)
plot(Auto2.svm.poly, Auto2, xx.mpg~ xx.acceleration)
5.SVM
library(MASS)
library(ISLR)
library(e1071)
library(LiblineaR)
set.seed(1)
#(a)
X1 <- c(3, 2, 4, 1, 2, 4, 4)
X2 <- c(4, 2, 4, 4, 1, 3, 1)
#Plot function graphical parameters: 2 = red, 4 = blue
Y <- c(2,2,2,2,4,4,4)
toy=data.frame(X1,X2,Y)
plot(toy$X1, toy$X2,col = Y)
#(b)
#The data frame 'toy' has two predictor dimensions so it may be possible to define the maximal margin hyperplane
#by defining the linear equation of a line sperating the two classes.
#By eye, we can see that observations 2 & 3 and 5 & 6 are logical support vectors for a hyperplane.
#The limits of the hyperplane margin are defined by lines that pass over observations 2 & 3 and 5 & 6.
#Both these lines have equal linear gradients:
# m = dX2/dX1 = 4-2 / 4-2 = 1 for observations 2 & 3
# m = dX2/dX1 = 3-1 / 4-2 = 1 for observations 5 & 6
#There for the linear gradient of the hyperplane line is also 1.
#The linear intercepts of these marging limits are as follows:
# y-4 = m(x-4) for observation 2 leading to a upper margin line defined by the linear equation X2=X1
# y-3 = m(x-4) for observation 6 leading to a lower margin line defined by the linear equation x2=X1-1
#There for the linear intercept of the hyperplane line is -1/2 = -0.5.
abline(a=-0.5,b=1, col = 3)
#There for the hyperplane line is defined by
#-0.5 + X1 - X2 = 0
#hyper plane coefficients: BETA0=-0.5, BETA1=1, BETA2=-1
#(c)
#classify to red if [-0.5 + X1 - X2 < 0] and classify to blue otherwise.
-0.5+toy$X1-toy$X2## [1] -1.5 -0.5 -0.5 -3.5 0.5 0.5 2.5#(d)
abline(a=0,b=1, col = 5)
abline(a=-1,b=1, col = 5)
#(e)
#The support vectors are observations 2 & 3 and 5 & 6.
#(e)
#As obvservation 7 is no a support vector, a slight movement such that it does not approach a margin will not impact
#the maximal margin hyper plane.
#(f)
#An example of a non optimal seperating hyper plane has the following equation and is plotted in red:
#[-1 + 1.2*X1 -X2 = 0]
abline(a=-1, b=1.2, col = 6)
#(g)
#An example of an additional Blue point [+] that would make the two classes no longer seperable by a hyperplane
#is shown at position (2, 3.5):
points(2,3.5, col = 4, pch = "+")
6. Hierarchical Clustering
library(MASS)
library(ISLR)
set.seed(1)
fix(USArrests)
?USArrests
#(a)
hc.complete <- hclust(dist(USArrests), method ="complete")
plot(hc.complete, main ='Complete', cex=0.9)
#(b)
cutree(hc.complete, k=3)## Alabama Alaska Arizona Arkansas California
## 1 1 1 2 1
## Colorado Connecticut Delaware Florida Georgia
## 2 3 1 1 2
## Hawaii Idaho Illinois Indiana Iowa
## 3 3 1 3 3
## Kansas Kentucky Louisiana Maine Maryland
## 3 3 1 3 1
## Massachusetts Michigan Minnesota Mississippi Missouri
## 2 1 3 1 2
## Montana Nebraska Nevada New Hampshire New Jersey
## 3 3 1 3 2
## New Mexico New York North Carolina North Dakota Ohio
## 1 1 1 3 3
## Oklahoma Oregon Pennsylvania Rhode Island South Carolina
## 2 2 3 2 1
## South Dakota Tennessee Texas Utah Vermont
## 3 2 2 3 3
## Virginia Washington West Virginia Wisconsin Wyoming
## 2 2 3 3 2#(c)
apply(USArrests , 2, mean )## Murder Assault UrbanPop Rape
## 7.788 170.760 65.540 21.232apply(USArrests , 2, var )## Murder Assault UrbanPop Rape
## 18.97047 6945.16571 209.51878 87.72916USArrests.scaled <- scale(USArrests, scale = TRUE)
apply(USArrests.scaled , 2, mean )## Murder Assault UrbanPop Rape
## -7.663087e-17 1.112408e-16 -4.332808e-16 8.942391e-17apply(USArrests.scaled , 2, var )## Murder Assault UrbanPop Rape
## 1 1 1 1hc.complete.scaled <- hclust(dist(USArrests.scaled), method ="complete")
plot(hc.complete.scaled, main ='Complete', cex=0.9)
cutree(hc.complete.scaled, k=3)## Alabama Alaska Arizona Arkansas California
## 1 1 2 3 2
## Colorado Connecticut Delaware Florida Georgia
## 2 3 3 2 1
## Hawaii Idaho Illinois Indiana Iowa
## 3 3 2 3 3
## Kansas Kentucky Louisiana Maine Maryland
## 3 3 1 3 2
## Massachusetts Michigan Minnesota Mississippi Missouri
## 3 2 3 1 3
## Montana Nebraska Nevada New Hampshire New Jersey
## 3 3 2 3 3
## New Mexico New York North Carolina North Dakota Ohio
## 2 2 1 3 3
## Oklahoma Oregon Pennsylvania Rhode Island South Carolina
## 3 3 3 3 1
## South Dakota Tennessee Texas Utah Vermont
## 3 1 2 3 3
## Virginia Washington West Virginia Wisconsin Wyoming
## 3 3 3 3 3#(d)
plot(hc.complete, main ='Complete', cex=0.9)
plot(hc.complete.scaled, main ='Complete', cex=0.9)
#Observing the denograms with unscaled variables on the left and scaled variable with SD = 1 on the right, the right hand
#plot has more gradual levels representing where all the parameters of the data set have equal impact on the clustering algorithm.
#Alaska stands out on the scaled dendrogram because of its low urban population yet comparable arrest rates to states with
#much higher urban populaions.7. PCA & K-Means Clustering
library(MASS)
library(ISLR)
set.seed(19)
#(a)
x = matrix(rnorm(60*50), ncol = 50)
x[1:20,]= x[1:20,]+2.4 #CLUSTER A
x[21:40,]= x[21:40,]+0.5 #CLuSTER B
x[41:60,]= x[41:60,]+1.2 #CLUSTER C
k <- c(rep('CLUSTER A', times=20), rep('CLUSTER B', times=20), rep('CLUSTER C', times=20))
k_col <- c(rep(2, times=20), rep(3, times=20), rep(4, times=20))
#(b)
pca.x <- prcomp(x)
pca.x## Standard deviations (1, .., p=50):
## [1] 5.70043345 1.73316723 1.68258007 1.64441414 1.59386890 1.55148592
## [7] 1.51289350 1.47090301 1.41745951 1.37640139 1.35034907 1.31500472
## [13] 1.27505467 1.24467602 1.14904670 1.12918122 1.09922696 1.05619019
## [19] 1.03675005 1.02587138 0.96064811 0.93903240 0.87976089 0.87048887
## [25] 0.83701095 0.81944756 0.79535198 0.77053575 0.71878544 0.70192274
## [31] 0.69333026 0.66626286 0.63229137 0.60818074 0.54963197 0.52269952
## [37] 0.48695590 0.46216902 0.44088277 0.40994136 0.35660038 0.33472843
## [43] 0.31936478 0.26555329 0.20859858 0.19359866 0.17547798 0.14425067
## [49] 0.08876469 0.06981792
##
## Rotation (n x k) = (50 x 50):
## PC1 PC2 PC3 PC4 PC5
## [1,] 0.14315202 0.020569318 -0.110037211 0.176223585 -0.013850384
## [2,] 0.12610636 -0.233011591 0.081330827 -0.190910191 0.164258986
## [3,] 0.19276192 0.144821992 -0.187176867 -0.191588589 -0.016114880
## [4,] 0.17508489 -0.104132981 -0.169353613 0.320467109 -0.175868346
## [5,] 0.12719038 -0.002095026 0.014385494 0.067962499 -0.131620600
## [6,] 0.16141878 0.201024647 0.222315842 -0.106961667 0.082123819
## [7,] 0.09119935 -0.010925516 0.146180415 -0.088137740 0.082365978
## [8,] 0.16303827 -0.159628827 0.072397828 -0.011547914 0.143577200
## [9,] 0.14297033 -0.036925263 -0.012993590 -0.027589134 -0.105775147
## [10,] 0.16432252 0.303688856 0.117536878 -0.187889346 0.021600310
## [11,] 0.12887360 -0.074900284 0.179853816 -0.191772256 -0.160226121
## [12,] 0.17269567 0.209227555 -0.026796362 0.238360376 -0.038284366
## [13,] 0.12129558 0.167127150 -0.266775155 -0.004101417 -0.045278392
## [14,] 0.17142903 0.037704170 0.109553781 0.265787845 -0.003459431
## [15,] 0.13633103 -0.081741864 0.076192438 0.154951185 0.233203829
## [16,] 0.14123974 0.057624887 -0.021150634 -0.124315319 -0.179463752
## [17,] 0.10895098 0.241837720 -0.046252580 -0.105773245 0.031728009
## [18,] 0.16407555 -0.145627590 0.138200881 -0.044637973 -0.114437734
## [19,] 0.14932051 -0.074479455 0.071822847 -0.172991286 -0.196662644
## [20,] 0.16638833 0.114761082 -0.129651853 0.036279546 -0.342219001
## [21,] 0.14918036 0.046131746 -0.033406536 0.193789785 0.133407808
## [22,] 0.14898155 -0.043202224 -0.106143983 -0.134896254 0.201739867
## [23,] 0.09638570 -0.002337495 0.060322740 0.122717682 0.051021264
## [24,] 0.14437935 0.004788681 0.106079545 -0.005330893 -0.089222467
## [25,] 0.14661390 -0.094135587 0.007600119 0.038628138 0.301591423
## [26,] 0.13671053 0.295459712 -0.201019355 -0.305116512 0.160077291
## [27,] 0.14322091 -0.253266509 0.196285103 0.136939847 -0.179864236
## [28,] 0.13050584 0.022718508 0.103478777 0.078671462 -0.090108871
## [29,] 0.17092806 -0.223966737 -0.113599841 -0.092763240 -0.071662490
## [30,] 0.12049760 0.011471902 0.077199840 -0.001897891 -0.214078316
## [31,] 0.09502931 0.238753990 -0.022808505 0.293204198 -0.038298084
## [32,] 0.12832383 0.146764038 0.330299087 0.175777727 0.335802577
## [33,] 0.10925551 -0.153282648 0.081203687 -0.079907024 -0.097412231
## [34,] 0.17023087 -0.029582867 0.107209463 -0.014883440 0.080463564
## [35,] 0.11466515 0.206276807 0.225824045 -0.046087585 -0.020266200
## [36,] 0.13421267 -0.196452780 -0.317732216 -0.164833378 0.103173203
## [37,] 0.14229398 0.161373039 0.091865952 0.080257529 0.006901402
## [38,] 0.16069967 -0.028831727 -0.107791233 0.057940769 0.136550947
## [39,] 0.15886617 -0.015258188 -0.184992063 0.111129929 0.137852289
## [40,] 0.15009341 -0.048557029 -0.111151232 0.059002523 -0.161052715
## [41,] 0.15241599 -0.146950623 -0.287055737 0.125131305 0.064642543
## [42,] 0.13081344 -0.050500085 -0.054157488 -0.097552782 0.202200268
## [43,] 0.11573324 -0.033674275 0.086352779 -0.144012178 0.053502621
## [44,] 0.12874929 0.161957178 -0.098335239 -0.102708915 -0.188653865
## [45,] 0.16650592 -0.253662511 0.197153207 -0.053836235 -0.059557400
## [46,] 0.10886248 0.033679002 -0.029160394 -0.065095683 0.036937932
## [47,] 0.10658892 -0.113050917 -0.127998117 -0.072286835 0.070884212
## [48,] 0.11929440 -0.053540654 -0.070949712 0.152964563 0.076258968
## [49,] 0.14503044 -0.002879098 0.115893453 -0.116778355 -0.093399551
## [50,] 0.09694950 -0.073549253 -0.049789417 -0.041166770 0.053646635
## PC6 PC7 PC8 PC9 PC10
## [1,] 0.1904537451 -0.2360331739 0.061576387 -0.002452703 -0.173595556
## [2,] 0.0694763492 0.0997616423 -0.067910245 -0.035521972 0.007116069
## [3,] -0.0774493344 0.0690004381 0.055008835 -0.088256359 0.112084477
## [4,] 0.0586929555 0.0380847369 -0.085682111 0.095303711 -0.099713299
## [5,] -0.1594777679 -0.0584208216 -0.235119532 0.054096132 -0.239552189
## [6,] 0.1727984341 -0.1138784188 -0.270278930 -0.028433311 0.035900749
## [7,] -0.0315285710 0.0999534916 -0.002548205 0.066786678 0.215746984
## [8,] -0.1982293758 -0.5148117619 -0.011954937 0.040656602 0.044462648
## [9,] -0.0359776054 0.0285142771 0.011661746 0.012539917 0.226372709
## [10,] 0.1255347954 0.0394735239 0.040895818 -0.343459593 -0.170131286
## [11,] 0.1319628050 0.0222503331 0.034749117 -0.041685613 0.105558768
## [12,] -0.1452298203 0.0427152036 0.015185430 -0.083439346 -0.284464236
## [13,] 0.0514924039 -0.0300728002 0.048459997 -0.089386022 0.040641306
## [14,] -0.2308469828 -0.0398915542 0.080778842 0.172287052 0.283531275
## [15,] 0.1963295996 0.1005885646 -0.026558178 -0.028063874 0.194186980
## [16,] -0.0833236661 -0.1420916460 0.233400634 0.166784501 0.016422236
## [17,] -0.1903803760 -0.0561881541 0.010645607 0.204719117 0.210220176
## [18,] 0.2823489885 0.0192829952 0.019750988 0.115648255 -0.025788603
## [19,] -0.0853912707 0.1931812116 0.228454725 0.109747133 0.165370024
## [20,] 0.1359490062 -0.0864227367 0.158455067 -0.327060383 0.222334798
## [21,] 0.0382571237 -0.0597317067 -0.079170042 0.112229801 0.059665795
## [22,] -0.1840315529 -0.1383081662 0.202036494 -0.168246262 -0.034894466
## [23,] -0.2610817390 0.2403475894 -0.267808132 -0.301096172 0.045357797
## [24,] 0.1804013049 0.0448094525 0.293381854 0.224288682 -0.169654795
## [25,] 0.0270141061 0.2655817077 0.094882672 -0.107849951 -0.046128289
## [26,] -0.1391046839 0.0445957911 -0.065232727 0.200172362 -0.139058829
## [27,] -0.0343057382 -0.0861631453 0.002893815 -0.170516662 0.017928189
## [28,] -0.0062290420 0.0978389005 -0.001971884 0.206563686 -0.076208645
## [29,] -0.0416538795 0.0686575611 -0.173193130 -0.055974764 -0.108300817
## [30,] 0.0020223111 -0.0791596731 0.084175481 0.001579757 -0.069901992
## [31,] -0.0071931678 0.2174398695 0.031811302 -0.036573681 0.151166321
## [32,] -0.0073700930 -0.0232574522 0.257054450 0.039375512 -0.114135671
## [33,] -0.2718846200 0.0772790888 -0.325386305 0.154269961 0.041345419
## [34,] 0.0547585475 -0.0176557302 0.080657495 0.083933715 -0.324843512
## [35,] 0.1051989418 -0.0336398340 -0.225529747 -0.099501336 0.260409493
## [36,] 0.0018984980 0.1365310402 0.189611040 -0.106737864 0.026960187
## [37,] -0.2043037047 -0.0201718092 0.036314791 -0.090986021 0.022430940
## [38,] 0.2728038609 -0.0527120981 -0.226633363 0.011255042 -0.024543472
## [39,] -0.0391764101 -0.0944281322 -0.061483307 0.139170101 0.052610873
## [40,] 0.1611130402 0.0551614358 -0.084705924 0.079570497 0.090657281
## [41,] 0.0281175666 0.0004370221 0.028338000 -0.072926583 0.019860947
## [42,] 0.0413360716 0.0944168642 0.027573357 0.206268521 0.145286743
## [43,] 0.2892060181 -0.1072242430 -0.094649524 0.089108587 0.060152552
## [44,] 0.0001172125 -0.0268541350 -0.173951793 0.218648345 -0.155420709
## [45,] -0.2588279252 -0.0848083296 0.106156329 -0.142990182 -0.095033999
## [46,] 0.0444695454 -0.2298308969 -0.115087457 -0.276629115 -0.031543674
## [47,] -0.0200712431 -0.0355755723 -0.202842019 0.006685687 -0.006848802
## [48,] 0.0591816770 0.2012899767 0.070642298 0.031946767 0.024590897
## [49,] -0.0599175756 0.3729005430 -0.041608581 -0.038913817 -0.248475990
## [50,] -0.0414515869 -0.1301194426 0.096569309 -0.062256318 0.030587075
## PC11 PC12 PC13 PC14 PC15
## [1,] 0.096241774 -0.144023688 0.3486490418 -0.02165015 -0.072884370
## [2,] 0.208572291 0.250220204 -0.1632291619 -0.39343672 -0.041443149
## [3,] 0.301141010 -0.065149629 -0.0367805107 -0.09811919 -0.278334067
## [4,] 0.085156657 0.271302724 0.2049740543 -0.12824101 0.185712784
## [5,] -0.165972518 -0.050683035 -0.2380655870 0.13264287 -0.114571519
## [6,] -0.009586066 0.098441865 -0.0977156091 0.08577415 -0.036655738
## [7,] -0.113568448 0.194428781 0.1109203772 0.09312707 0.019318064
## [8,] -0.032769937 -0.059280963 -0.0524929003 -0.26057392 -0.077614753
## [9,] 0.011849542 0.205159879 0.0767659740 -0.01583086 0.002457592
## [10,] 0.123254405 0.134593177 0.0794626873 -0.09660302 -0.017513424
## [11,] 0.034966877 -0.345788008 0.2555637810 0.09346358 0.334043935
## [12,] 0.030972523 0.030734144 -0.0503256114 0.08540928 -0.102491001
## [13,] -0.098696789 -0.116329869 0.0470878191 0.04301613 -0.109764912
## [14,] -0.099159988 -0.097942569 -0.1977061311 0.13646308 0.040014358
## [15,] -0.137410206 0.100659480 -0.0022379293 0.14663716 -0.193708023
## [16,] 0.095369169 -0.072476697 0.1187171284 -0.07993014 0.024091178
## [17,] -0.100263330 0.114796250 0.1349740586 0.03554168 -0.098908118
## [18,] -0.009030159 -0.053377513 -0.0251707225 0.14117684 -0.286445219
## [19,] 0.063437575 -0.020288534 0.0717641971 0.19852343 -0.222421918
## [20,] -0.163660115 -0.047128064 -0.2727646548 -0.20570426 0.037845803
## [21,] 0.199915065 -0.310308763 -0.1618238645 0.15171767 -0.010048096
## [22,] -0.079881383 0.153031491 -0.0561460483 0.01924158 -0.027035003
## [23,] -0.049885878 0.028108365 0.1347798222 0.07229026 0.014668947
## [24,] 0.106418359 0.099394335 -0.0512159656 0.03011886 0.033564915
## [25,] 0.174513227 -0.120321504 -0.1010485214 0.11663825 0.096614497
## [26,] -0.052129755 -0.077018188 -0.0320306972 -0.08514803 0.211655455
## [27,] 0.048667208 -0.112743492 -0.1760647190 -0.05398538 0.125610190
## [28,] -0.117541813 0.151148156 0.1924439841 0.10988441 -0.177261636
## [29,] 0.185868818 -0.094403516 -0.0880157069 0.13958062 0.118306642
## [30,] 0.038241820 0.299460397 -0.2644111260 0.16944709 0.224460889
## [31,] -0.176855175 -0.211087582 -0.0693443971 -0.33733967 0.014196271
## [32,] -0.131063335 -0.020409823 0.1340394904 -0.09372747 0.049464022
## [33,] -0.102514438 -0.028220688 0.1614640054 -0.20150586 0.022743877
## [34,] -0.244910553 -0.102550399 -0.0527780637 -0.09822019 -0.024767363
## [35,] 0.142684816 -0.169967444 0.1573967326 -0.01459822 0.038282306
## [36,] -0.403536044 -0.063261854 0.0829200021 0.07984549 0.189299958
## [37,] 0.204890983 0.204579923 -0.0572148873 0.17031404 0.233050690
## [38,] -0.056237386 0.044083146 0.0126329447 0.03337905 0.175434223
## [39,] 0.305760504 0.048734994 0.0881050987 -0.13487203 -0.053463232
## [40,] -0.116876426 0.245557402 0.0342562102 -0.15254071 -0.076530573
## [41,] 0.148193319 -0.055301715 0.1681807117 0.17293906 -0.055136063
## [42,] 0.023382864 0.007840987 -0.1802880708 0.06589222 0.252001303
## [43,] -0.131253166 -0.067031934 -0.0831491534 -0.08746111 0.017019524
## [44,] -0.053348389 -0.053127123 -0.0557832797 0.06843266 0.171159447
## [45,] -0.016127731 -0.081827990 0.2214588086 -0.07516022 0.038544727
## [46,] -0.208214162 0.155907816 0.1706436705 0.23837294 0.014871879
## [47,] -0.108264010 -0.091454474 -0.0947399413 0.02599144 -0.241645231
## [48,] -0.053351170 0.047027716 0.0006445563 -0.19927045 0.103491572
## [49,] -0.030728184 -0.121110249 -0.0591768829 -0.09856165 -0.200543753
## [50,] 0.002675002 0.008154446 -0.1706696746 0.12857345 -0.182550914
## PC16 PC17 PC18 PC19 PC20
## [1,] 0.2796445940 0.055601694 -0.0338521415 -0.182615491 0.109750828
## [2,] -0.0886176265 0.283246165 0.0708958712 0.038658437 0.007117845
## [3,] 0.2055237241 -0.189396715 0.0219705236 0.070751717 -0.074466537
## [4,] 0.0704168434 -0.087934485 0.0103626456 -0.080279387 0.018789983
## [5,] 0.1436213422 0.107625353 0.2019408700 0.093768075 -0.142271427
## [6,] -0.0650755512 -0.060631900 -0.1053678680 0.197992084 0.292037287
## [7,] -0.0041811431 -0.263355581 -0.3548437549 -0.085605434 -0.133747658
## [8,] 0.2196456111 -0.104820567 0.0951148615 -0.068353094 -0.049726256
## [9,] -0.0182554336 0.008774002 0.0010885445 0.149295583 0.120096102
## [10,] 0.1530816336 -0.088338092 0.1141639897 0.062551329 0.064179424
## [11,] -0.0002372424 0.093905360 0.1371369214 -0.008052025 -0.023254481
## [12,] -0.0553793317 0.213514056 0.0600232083 0.055582690 -0.158820114
## [13,] -0.0855341019 0.094214609 -0.1097249369 0.099715468 0.165612018
## [14,] 0.1244730872 0.041550844 0.0955446290 -0.068705463 0.153201237
## [15,] 0.1379838735 0.179856120 0.0204136469 -0.113891180 0.054398071
## [16,] -0.1341698758 0.158788371 -0.0341238966 0.015692204 -0.016310058
## [17,] -0.1992020120 0.100626967 0.2114616327 -0.123272918 0.059322905
## [18,] 0.0648994327 0.170149556 -0.3296569259 0.145139614 -0.241550645
## [19,] 0.0523743288 -0.144762788 0.3470028086 0.018348606 0.227178046
## [20,] 0.0287696521 0.034211040 -0.2053420664 0.059141701 0.040731312
## [21,] -0.2409005896 -0.138056989 -0.2405552841 0.076350159 0.120982277
## [22,] -0.0204698420 -0.067104349 -0.0984379037 -0.127947773 0.162769605
## [23,] 0.0674627780 0.082952584 -0.0170814152 0.033117732 0.021325731
## [24,] -0.0489034061 -0.179294182 0.0009124662 -0.203681680 -0.169020081
## [25,] -0.2453280705 0.055939146 0.0071036687 -0.208190392 0.118632170
## [26,] -0.1094730686 -0.029472297 -0.1295771919 -0.043681082 0.024181721
## [27,] -0.2079099106 -0.244876675 0.0676349246 -0.008687854 0.176992193
## [28,] -0.0377794920 -0.007072280 -0.1033316377 0.140766803 0.151375935
## [29,] 0.0739835923 -0.060435966 0.0156818140 0.036236768 0.075200993
## [30,] 0.2517129838 0.057171965 -0.1412826727 -0.109578110 0.084389659
## [31,] -0.0727895306 -0.143786070 0.0108740682 0.068192512 -0.218882329
## [32,] 0.1010927467 -0.167517093 -0.0079422482 0.086286748 -0.080927047
## [33,] 0.0704725346 0.069214718 -0.1710290066 0.068399537 0.054193275
## [34,] -0.0549050711 0.168797791 0.1221763118 0.219991226 0.267979030
## [35,] 0.1259397577 0.204285473 0.1195909006 -0.205549407 -0.158661159
## [36,] 0.1929209074 0.076511853 -0.0149537147 0.042242425 -0.002451679
## [37,] -0.0306268428 0.152974689 -0.0692061145 -0.096627238 -0.133388997
## [38,] -0.0100683218 -0.284162685 0.3426644906 0.076303017 -0.002299501
## [39,] -0.0140456665 0.033833442 -0.1025890676 0.163505435 0.049064123
## [40,] -0.3000971317 -0.126373422 0.1954916100 -0.004272270 -0.132880332
## [41,] -0.0546007927 0.050744656 -0.0314258002 0.146245949 -0.090309992
## [42,] 0.2554664205 -0.055422660 0.0551445139 0.352826621 -0.320508018
## [43,] -0.1214281275 0.180511578 -0.0656281959 -0.140402561 0.010339085
## [44,] -0.0491002813 0.016416192 -0.0146826214 -0.179151418 -0.061961246
## [45,] -0.2261455734 -0.015888476 -0.1273110206 0.072691368 -0.195483289
## [46,] -0.2348009219 -0.107485002 0.0001510426 0.004486426 -0.185729272
## [47,] 0.0743477082 -0.200706833 -0.0860847425 -0.453030561 -0.047554994
## [48,] 0.0091524486 0.236943876 0.0040464013 -0.272768717 0.080044403
## [49,] 0.0683903749 -0.166986167 -0.0422752729 -0.136537203 -0.058131378
## [50,] -0.1964373370 0.163729526 0.2197221194 -0.027364655 -0.314466832
## PC21 PC22 PC23 PC24 PC25
## [1,] 0.134775051 -0.144269717 -0.044616669 -0.163477843 0.037333659
## [2,] 0.068723628 0.103571100 0.033136816 -0.021869522 -0.135159150
## [3,] 0.055999429 -0.047825567 -0.181042709 0.000433723 0.183546487
## [4,] -0.057774272 -0.045774325 -0.077618624 0.106838211 0.045184006
## [5,] 0.031996760 0.276078461 0.061487447 0.080634387 -0.103625726
## [6,] 0.039342580 0.049616356 -0.028464523 -0.106782978 -0.268269234
## [7,] 0.196783839 0.060815533 0.295776812 -0.027815915 0.265369249
## [8,] 0.107446157 -0.060580134 0.146123910 -0.084490313 0.074431661
## [9,] 0.190802797 0.007573996 -0.129199923 -0.188219163 -0.098636468
## [10,] -0.034182065 0.099527925 0.055174083 0.249318200 0.124051356
## [11,] -0.147251277 0.011680007 -0.109590316 -0.102938854 -0.154745726
## [12,] -0.077796885 -0.232472096 0.213735580 -0.091742095 0.074982900
## [13,] -0.258187732 -0.020436363 0.336588829 0.123766345 0.291249016
## [14,] 0.094389425 0.006015762 0.022968051 0.192415245 -0.016179095
## [15,] -0.356070066 -0.040796044 0.040738371 -0.400956449 0.039733110
## [16,] -0.064161669 0.020912178 0.044095767 -0.080282605 -0.195753247
## [17,] 0.025080146 0.010410168 -0.008383990 0.096710761 0.004929064
## [18,] 0.102212466 -0.073601456 -0.179142024 0.331201080 0.090629650
## [19,] -0.005430026 -0.025630460 -0.021458606 -0.001173090 0.043883100
## [20,] -0.048099439 0.117657111 -0.107880338 -0.107002459 -0.072394341
## [21,] -0.033658184 -0.047197440 0.041541468 -0.075491815 -0.145981723
## [22,] -0.289107254 -0.057998241 0.130616671 0.176172398 -0.292238135
## [23,] -0.135690839 0.242923859 -0.213242811 -0.023878991 0.119458377
## [24,] -0.119244329 0.443396588 0.193138774 -0.177477969 -0.005908400
## [25,] 0.150317311 0.038399314 0.076208892 0.136370600 0.104786535
## [26,] 0.065036966 0.001079783 -0.175663700 -0.133636772 0.133061758
## [27,] -0.103817952 -0.200321169 -0.133539809 0.081119903 0.279422311
## [28,] 0.032610246 0.075225498 -0.162485735 0.103672922 -0.047072310
## [29,] 0.259404261 -0.056800702 0.253511122 0.144674335 -0.167120085
## [30,] 0.022393867 0.011655097 0.053267967 -0.001962173 -0.020253128
## [31,] 0.180795825 0.137608127 0.051530962 -0.036084016 -0.126279141
## [32,] 0.053728141 -0.091980628 -0.165452835 0.184184881 -0.183635052
## [33,] -0.079547316 -0.085158023 0.146867697 -0.051267937 0.031912991
## [34,] 0.143981865 0.119204477 0.045150513 -0.158694204 0.237254644
## [35,] 0.053509930 0.041520089 0.271396655 0.147142545 -0.041415053
## [36,] 0.235860437 -0.039821905 -0.017784076 0.014862645 -0.037491182
## [37,] 0.213572389 -0.146584243 -0.163214750 -0.285564608 0.152081817
## [38,] -0.027663502 0.101004524 -0.056292113 -0.062378287 0.107548661
## [39,] -0.151486544 0.067893459 -0.146982009 0.094552213 -0.035688961
## [40,] 0.043552083 -0.230174153 0.173089870 0.025710762 -0.029942380
## [41,] 0.157810073 0.208717792 0.117741188 -0.105759619 -0.141318861
## [42,] -0.337487226 -0.154901344 0.030599023 0.028004991 0.045414636
## [43,] 0.068202678 -0.112960523 -0.049662901 0.026309659 0.213582434
## [44,] -0.164212988 -0.005167821 -0.130230871 0.139731797 -0.008207584
## [45,] -0.179082992 0.200256230 -0.009403441 -0.005376599 0.111018779
## [46,] 0.021643232 -0.139672700 -0.035676736 0.051124985 -0.191576920
## [47,] -0.055048904 0.212856424 -0.218526235 -0.015698973 -0.036657127
## [48,] -0.071945119 -0.040522228 -0.107346846 0.235132792 -0.093803119
## [49,] -0.047380021 -0.385173027 0.041599296 -0.233776386 -0.238698805
## [50,] 0.055066113 -0.052575093 -0.224554351 -0.020686146 0.041204920
## PC26 PC27 PC28 PC29 PC30
## [1,] 0.025367963 0.0299650628 0.1871501825 -0.203545990 -0.1439739810
## [2,] -0.026468725 0.0481811725 0.1473375393 0.064221896 0.0115245126
## [3,] 0.179360071 -0.2557520213 -0.2435446400 0.023025231 -0.2052952858
## [4,] -0.222418483 0.1598853672 0.0023668314 0.111506089 0.0001979637
## [5,] -0.231647953 0.0730375856 -0.0620220108 -0.054956536 -0.1387723992
## [6,] -0.012509613 0.0133065331 -0.0757919423 -0.076385009 -0.0366922570
## [7,] -0.057961467 -0.1477203287 0.0870570260 0.074252695 -0.2582008101
## [8,] -0.243293566 -0.0870003513 -0.0606707815 -0.083123721 0.1667624894
## [9,] 0.162765449 -0.1199301645 0.3539049124 -0.050277645 -0.0116482529
## [10,] 0.009719559 -0.0661012824 0.0662009461 -0.045343266 0.2650075300
## [11,] -0.120010544 -0.0679859983 0.1662967865 0.127092203 0.0524297498
## [12,] 0.136753683 -0.0908540928 0.1989452155 -0.003419889 -0.1166289896
## [13,] 0.054012382 0.2540746695 0.0510352794 0.082987275 0.1281455238
## [14,] 0.033579936 0.0245615373 -0.0924013444 -0.218459174 0.1509471908
## [15,] -0.117699300 -0.1102853846 -0.2078320888 0.154615748 -0.0969929478
## [16,] -0.055315954 -0.0601512196 -0.1664849572 0.023934017 -0.2845379619
## [17,] -0.051807505 0.1211793369 0.1865461808 0.191060135 -0.1632467215
## [18,] -0.254179659 0.1191486431 0.0432764288 0.187444045 0.1035911316
## [19,] -0.123870967 0.1615345453 -0.0912595787 0.002954691 0.0337063459
## [20,] -0.056490567 0.1365167709 -0.0143596326 0.018336395 -0.0478474695
## [21,] 0.083293927 0.0062801475 0.1497965811 -0.088906125 0.0808526298
## [22,] 0.005804453 0.0829880366 0.0567081176 0.138715035 -0.0119358782
## [23,] -0.087305038 0.0002071189 -0.0686014524 -0.106544860 -0.1821136966
## [24,] 0.183514233 0.0187857975 -0.1096080875 -0.021706778 0.2019566032
## [25,] -0.178958027 -0.1370068844 0.0043290505 -0.177205967 0.0004602244
## [26,] -0.377894446 0.0674759854 -0.0817860135 -0.086038168 0.1045211571
## [27,] -0.060041888 -0.2073383598 0.0235427937 0.189257195 0.0301236083
## [28,] 0.130269028 -0.0477515873 0.0545977683 -0.244268047 0.1657371881
## [29,] 0.174013566 0.2027075674 -0.0634603669 0.189101406 -0.3096960029
## [30,] 0.013287558 -0.1531293392 0.0100837553 -0.121641760 -0.0521058253
## [31,] 0.012649513 -0.0278506506 0.0250464458 0.076824091 -0.0043716967
## [32,] 0.094591323 0.1108380124 -0.1205713469 0.283164920 -0.1431019170
## [33,] 0.308380653 -0.0357237853 -0.1876093418 0.189535450 0.3430269447
## [34,] 0.016293614 -0.2114297701 0.1623394347 0.200565654 -0.0534809803
## [35,] 0.015475085 -0.0444721744 0.0827725718 -0.049563573 0.0825328926
## [36,] 0.141974323 -0.0335459251 0.0007180346 -0.108775714 0.1039086511
## [37,] 0.042261787 0.3286492397 -0.0339597400 0.200235177 0.2030216899
## [38,] 0.184576517 0.2552260257 -0.0797403151 0.003364961 -0.0307765939
## [39,] 0.026012999 -0.0273519839 -0.0280109247 -0.104306504 -0.0014692534
## [40,] -0.160976325 -0.0725345495 0.0235207881 -0.156225762 -0.0153873729
## [41,] -0.154701540 -0.1646408054 -0.2138892686 0.192823771 0.2049779347
## [42,] -0.025224591 -0.0212701927 0.2547304990 -0.132711383 -0.0399473477
## [43,] 0.204570271 0.2123094129 -0.2881693454 -0.196220238 -0.1801007431
## [44,] 0.128437173 -0.3511147559 -0.0748105858 0.152517514 -0.0701092522
## [45,] 0.079491167 0.1856489728 0.0260689209 -0.319507260 -0.1681598880
## [46,] -0.026353498 -0.1906506371 -0.1378937127 -0.071493483 0.0900918633
## [47,] 0.074449351 0.0643286331 0.3959127762 0.144022609 0.0322774369
## [48,] 0.079319047 -0.1268027937 -0.1135238063 -0.063764251 0.0041153531
## [49,] -0.076235539 0.1234068952 -0.0321646317 -0.155172490 0.0933246930
## [50,] 0.255249211 -0.0289501121 0.0129539035 0.003194214 0.1345008387
## PC31 PC32 PC33 PC34 PC35
## [1,] -0.185966454 0.0587712208 0.154023000 0.2352545267 -0.026204167
## [2,] 0.124000494 -0.0690460558 0.087952057 0.0450448245 0.021481759
## [3,] 0.062918504 -0.3226853992 0.010401240 -0.0722947620 0.012317698
## [4,] -0.253683689 -0.3416264826 -0.073270901 0.0007493148 0.028608853
## [5,] -0.155444085 -0.0385682921 0.160608122 0.1974336513 -0.151102325
## [6,] -0.157562402 -0.1524306098 0.065650942 -0.0385801348 0.010924734
## [7,] -0.120131905 -0.0112489245 0.008017331 0.1436498378 -0.175012450
## [8,] 0.219743627 0.0685144832 0.001163075 0.0801186767 0.352830234
## [9,] -0.023886320 0.0519690063 0.145200205 0.0355546575 0.181082022
## [10,] -0.052007866 0.1076485961 -0.050506528 0.1891379465 -0.165775342
## [11,] 0.019469780 -0.0823537738 -0.050061239 0.0977812545 -0.013836979
## [12,] 0.129362626 0.0930301550 -0.299876083 -0.1659192074 0.007847047
## [13,] 0.107430646 -0.0221531994 0.313555693 0.0548534897 0.074379205
## [14,] -0.070789672 -0.1999462932 -0.081217070 -0.1416619587 -0.161859651
## [15,] -0.108616623 0.0742753493 -0.011516352 0.0084739923 -0.021729287
## [16,] 0.118118183 0.2397789399 -0.257878970 0.1486602729 -0.230452860
## [17,] 0.041423945 -0.0962868155 0.203423088 0.1383780398 -0.010139604
## [18,] 0.041962656 0.1119344732 -0.105995594 -0.0105091389 0.159032297
## [19,] -0.045172253 0.2668938371 0.038847997 -0.1878588348 0.137759167
## [20,] 0.021589811 -0.0535495080 -0.074639099 -0.0487502044 -0.079078288
## [21,] 0.155536344 -0.1509180670 -0.087587689 0.1568947477 0.068776736
## [22,] -0.131242826 -0.0358640826 -0.190321128 0.1311802029 0.131016187
## [23,] 0.166820454 -0.0114154909 -0.224694630 0.1974451418 0.311852676
## [24,] -0.127004567 -0.1271525319 -0.106967193 -0.0019097412 0.257091486
## [25,] -0.060305941 0.1595630983 0.045794035 0.1102615270 0.032212782
## [26,] -0.077081587 0.0254833616 -0.121503336 -0.0480109424 -0.108352585
## [27,] -0.073866125 0.0370361435 0.011649133 0.0947337668 -0.118181429
## [28,] 0.202467177 0.0145415045 -0.094828646 0.1567103282 0.018740195
## [29,] -0.111171682 0.0616009823 -0.055385158 -0.0458540428 0.142739692
## [30,] 0.209154668 0.2313521084 0.168136696 0.0058555552 -0.168818517
## [31,] -0.185384829 0.3428478116 0.132557219 0.0665947078 0.190413633
## [32,] 0.179707801 -0.0565990822 0.053177451 -0.0183661687 -0.104172681
## [33,] -0.109629520 0.1393824800 -0.051844776 0.1275819775 -0.167367907
## [34,] -0.098692102 -0.1342991262 -0.037944292 -0.2955076518 -0.001600701
## [35,] -0.024563814 -0.1126517366 -0.144004960 -0.2293008722 -0.045964184
## [36,] -0.016715681 -0.0776167361 -0.199953673 -0.0026803150 0.042105696
## [37,] 0.057262114 0.0262030646 0.056530824 -0.0875489277 0.085672811
## [38,] 0.265449855 0.2061999472 -0.070603842 0.0572433112 -0.147012626
## [39,] -0.297722904 0.3224058877 -0.062120762 -0.2666381922 -0.071374995
## [40,] 0.170211064 -0.0985901048 -0.213531762 -0.0202064179 -0.057897338
## [41,] 0.269162599 -0.0968249362 0.220422072 -0.0086087062 -0.230464434
## [42,] -0.034250853 -0.0147401790 0.125353268 -0.0090589121 0.008196382
## [43,] 0.003447964 0.0093480218 0.001437967 0.0337795303 0.078963536
## [44,] 0.088811226 -0.0006105085 0.186053571 0.0258619003 0.303732071
## [45,] 0.026284471 -0.0917414380 0.236523721 -0.2214995016 -0.091018276
## [46,] -0.149459026 0.1114370314 0.152588122 -0.2828869525 0.117400178
## [47,] 0.125087178 0.1012452051 -0.072342433 -0.2442070633 -0.213789879
## [48,] 0.099478127 -0.0011752173 0.242268215 -0.0886808627 -0.030040566
## [49,] 0.019437250 -0.0896457355 0.192129623 0.0207463346 -0.067752940
## [50,] -0.304480641 -0.0657439752 -0.023850841 0.3467536576 -0.144194827
## PC36 PC37 PC38 PC39 PC40
## [1,] -0.113013692 -0.013635731 0.09217627 0.190039572 0.1062860544
## [2,] 0.118544630 -0.116581952 0.02403489 0.167886756 -0.0009604968
## [3,] -0.040744755 -0.153936274 -0.20230334 -0.046370925 0.1369202366
## [4,] 0.019507809 -0.082922504 0.01328199 -0.254467211 -0.1402474466
## [5,] -0.242201752 0.112875025 -0.08139808 -0.003409105 0.0046964470
## [6,] 0.113870008 0.105007827 0.13755287 0.197601944 0.1871432733
## [7,] -0.039781642 0.084611873 -0.05443410 0.094552084 0.1364491148
## [8,] 0.087387385 0.050684770 -0.06915648 -0.050695877 0.0235304060
## [9,] -0.181560650 -0.042673178 0.27882739 -0.001683822 -0.1518375648
## [10,] 0.040382177 -0.059985250 0.09175950 0.009493088 -0.3583843870
## [11,] 0.141459545 -0.266333926 -0.19180723 -0.093691214 0.2982573861
## [12,] 0.148729831 0.074520337 0.22392174 0.025806641 0.2851843914
## [13,] 0.019450551 -0.202154884 -0.02040641 0.093975467 0.0120330135
## [14,] 0.174535727 -0.268545161 0.13096323 0.167243612 -0.0797643375
## [15,] 0.140890457 -0.072393973 -0.16397571 0.138378646 -0.2772527864
## [16,] -0.131153854 -0.007107414 0.06582871 0.034428000 -0.2849406887
## [17,] 0.355011128 0.169913381 0.04389148 -0.221897471 0.0662587400
## [18,] 0.067862245 -0.067313657 0.06180857 0.125561308 0.0647318918
## [19,] -0.227807321 0.033380228 0.12496315 0.009400515 0.0506958420
## [20,] -0.054535841 0.214241729 -0.03521927 -0.118242668 0.0639191546
## [21,] -0.156538989 -0.100671487 -0.03120757 -0.061556798 -0.1943963562
## [22,] -0.201411193 0.067541234 -0.14770706 -0.031058946 0.1573872849
## [23,] -0.031809111 -0.097563070 0.22952378 -0.059707167 0.0665239548
## [24,] 0.046627885 0.039306499 0.18295749 0.022564752 0.1385934552
## [25,] -0.029782692 0.026565224 -0.07593412 -0.250676291 -0.0132750235
## [26,] -0.033162604 -0.112642753 0.09227188 0.210395897 0.1053872362
## [27,] 0.084085630 0.298189435 0.13388570 0.146980280 0.0368934660
## [28,] 0.017706953 0.216153079 -0.47517442 -0.050702989 0.0329790809
## [29,] 0.098091029 -0.092023967 -0.10559934 0.117367970 -0.0888917560
## [30,] 0.164696156 -0.285898318 -0.06779599 -0.228521795 0.1142755972
## [31,] 0.084868827 -0.134049493 -0.14425716 -0.007787922 0.0073646424
## [32,] -0.049973106 -0.074381649 0.14642554 -0.049958847 -0.0495417293
## [33,] -0.147419922 -0.101641374 0.09534540 -0.123605712 0.1055643685
## [34,] -0.165388356 -0.070075909 -0.16591523 -0.177994176 -0.0523615288
## [35,] -0.128890331 0.272000997 -0.09063547 -0.018990467 -0.0698617910
## [36,] 0.244555414 0.206414611 0.03383012 0.102487375 -0.1440422509
## [37,] -0.173765794 0.122209016 -0.24937079 0.088320994 -0.1082278622
## [38,] 0.040107531 0.049604924 -0.10865709 0.114080793 0.1421357704
## [39,] 0.257610260 0.148884046 -0.08588752 -0.114068699 0.0511252769
## [40,] -0.126036330 -0.184196479 -0.07009040 0.116007922 -0.0433308533
## [41,] 0.008384181 0.177381498 0.23084388 -0.116788690 0.0431519429
## [42,] -0.191978020 0.123434747 0.08197953 -0.114293898 -0.0306562983
## [43,] -0.063951572 0.041618486 0.20424973 -0.405384828 -0.0314683768
## [44,] 0.087469165 0.135660956 0.00344335 0.180795954 -0.2824816925
## [45,] 0.093234999 -0.026639697 -0.08095032 0.169048914 -0.1440442005
## [46,] -0.098236984 -0.189499310 0.01250810 -0.119613798 0.0222531145
## [47,] -0.059347862 -0.089845748 0.07037844 -0.071590974 -0.0864479137
## [48,] -0.301721885 0.077777024 -0.01878256 0.257385667 0.2738477690
## [49,] 0.185584281 0.133937209 0.01624358 -0.141018243 -0.0562864337
## [50,] 0.022771885 -0.110529012 -0.02872197 0.028976111 0.0685660610
## PC41 PC42 PC43 PC44 PC45
## [1,] -0.053532344 -0.210610375 -0.0351752450 0.188214797 0.102981260
## [2,] 0.067677687 0.008622570 -0.0075606163 0.096417905 0.176078738
## [3,] 0.071045605 0.130513502 0.0103526120 0.111603513 -0.198168374
## [4,] 0.204994465 0.158386575 -0.0810254454 -0.028344489 -0.037544400
## [5,] -0.065824306 0.284249638 -0.0113005929 -0.103424054 -0.120589668
## [6,] -0.066587237 0.103238314 -0.3200999122 0.102892726 -0.151461705
## [7,] -0.040355961 -0.027900699 -0.0008583171 -0.210340899 0.149245357
## [8,] 0.022081691 0.123524291 0.0089184515 -0.115473517 0.075789984
## [9,] 0.276215189 0.087381781 0.1079463864 -0.213834302 -0.226502499
## [10,] -0.175736995 0.012881988 0.2051275052 -0.210056270 -0.028992135
## [11,] -0.111888125 0.091224155 0.0043722850 -0.258781990 -0.058247496
## [12,] 0.151579126 0.166336195 0.0256465604 -0.229854274 -0.042259408
## [13,] -0.003117017 0.152048241 -0.1661047905 0.108637611 0.044205139
## [14,] 0.087473158 -0.274933484 0.1473364127 -0.003094131 -0.038574600
## [15,] 0.071116432 0.093360014 0.1274229457 -0.097674707 -0.008136812
## [16,] 0.047810336 0.094479658 0.0355539955 0.307116273 -0.107330762
## [17,] -0.092124437 0.027587461 0.2274570185 0.198446465 0.017456714
## [18,] 0.105901783 -0.075097751 0.0593547060 0.108057744 -0.180498499
## [19,] -0.028845103 0.040299642 -0.1941715501 -0.133489124 0.185225986
## [20,] 0.181059670 -0.070017423 0.0230705158 -0.072131291 0.402233711
## [21,] -0.142876728 0.200590053 0.0903593481 -0.053427101 0.108777105
## [22,] 0.045820920 -0.335594204 -0.0253846199 -0.084848162 -0.329278041
## [23,] -0.204704701 -0.175001645 -0.0216204603 0.184881392 0.140516562
## [24,] -0.079033825 0.081000764 0.1229381704 0.172886254 0.035599093
## [25,] 0.333891133 0.078196486 -0.2398908587 0.110006425 0.010873552
## [26,] 0.134418461 -0.009526182 0.1558366914 -0.078573261 0.082060512
## [27,] -0.110397596 0.085545702 0.1225721663 0.194428357 -0.107866838
## [28,] 0.071847118 0.006415378 0.1579184052 -0.007472906 0.123727232
## [29,] -0.068668720 -0.036498340 0.2691871347 -0.056196254 0.219594062
## [30,] -0.075939235 -0.013625100 -0.1491369261 0.118151309 -0.041530255
## [31,] -0.068008060 -0.082121527 -0.0164438444 0.051115324 -0.248262623
## [32,] 0.020038568 0.120647161 -0.2165434903 -0.057216667 0.230256621
## [33,] 0.109104587 0.043548125 -0.0766981839 0.101493345 0.054656412
## [34,] -0.094543598 -0.245019775 0.0342528697 0.114214462 0.031999383
## [35,] 0.131396740 -0.044095441 -0.1018541563 0.150613586 0.014745354
## [36,] -0.142111134 0.284451759 -0.1845048393 0.109655277 -0.091406634
## [37,] -0.220641610 -0.012279540 -0.0668962194 0.053485575 -0.127173002
## [38,] 0.321100370 -0.122949767 0.0544983233 0.054117482 -0.113196274
## [39,] -0.262083248 -0.001117219 -0.1078637728 -0.132553792 0.094773522
## [40,] -0.332884169 -0.113573338 -0.2381514791 -0.019805918 0.007990628
## [41,] -0.078453091 -0.282756911 0.0179709073 -0.142391372 -0.083648167
## [42,] -0.068391573 -0.096999948 0.0754006239 0.186025657 0.064250615
## [43,] -0.163784546 -0.056733516 0.0869985391 -0.249578617 -0.190456073
## [44,] 0.109363451 -0.228886407 -0.2577474302 -0.178569968 0.115063151
## [45,] 0.090468436 -0.042187929 -0.0684979762 -0.094133782 -0.128272498
## [46,] 0.067564037 0.112919940 0.2703317873 0.235180404 0.172664099
## [47,] -0.107717699 0.082910245 -0.0725515921 0.038428164 -0.036440286
## [48,] -0.160190968 0.185623058 0.2514272284 -0.081652531 0.006723520
## [49,] 0.013281793 -0.157649250 0.0675264668 0.049904304 0.004325586
## [50,] 0.033667643 -0.088438913 -0.1688821446 -0.069464701 0.228687721
## PC46 PC47 PC48 PC49 PC50
## [1,] -0.211152765 -0.037502977 -0.130751686 0.009228641 0.111816555
## [2,] 0.061283694 -0.184792831 -0.422257331 0.073142515 0.193676012
## [3,] -0.102783475 -0.023110172 -0.067310386 -0.024832499 0.028895396
## [4,] 0.202494862 -0.037323340 0.061690014 0.232402002 0.062710618
## [5,] -0.102519022 -0.023900289 -0.247365007 -0.190617493 -0.062353760
## [6,] 0.032661129 -0.139202391 0.325455462 0.153384514 0.055034702
## [7,] 0.189378937 -0.182385657 -0.130186993 0.090387808 -0.163477202
## [8,] 0.087819657 -0.047150813 0.216480336 0.036793158 -0.087711959
## [9,] 0.065173887 0.174669320 0.040392729 -0.341081590 -0.143162062
## [10,] -0.113988149 -0.080379422 0.101765909 0.134545036 -0.051449198
## [11,] -0.001086310 -0.179031633 -0.026311378 -0.110532955 -0.036126920
## [12,] -0.041924138 -0.120610996 -0.012644136 0.112104159 0.126210027
## [13,] 0.283871690 -0.062442876 0.008394039 -0.266649278 0.005028121
## [14,] -0.004558391 -0.278120641 -0.091157533 -0.091324760 -0.043094750
## [15,] -0.120243173 0.140449936 0.040958832 -0.024916900 0.099110725
## [16,] 0.250027234 -0.226295828 0.112905080 0.021895662 -0.095108259
## [17,] -0.273485316 0.119574944 0.043837992 0.122640773 -0.098231609
## [18,] -0.100595112 0.079939777 0.016773535 0.060081686 -0.187309522
## [19,] 0.031938298 0.021862025 -0.132355867 0.298599168 0.113541484
## [20,] -0.222234779 -0.052327381 0.049957445 -0.018984351 -0.070115544
## [21,] -0.050825453 0.198624331 -0.245542774 0.330849621 -0.135216869
## [22,] -0.006161111 0.081310409 -0.140733270 0.003563248 0.041127632
## [23,] 0.137611003 0.043951374 0.009861859 -0.042196039 -0.093466610
## [24,] -0.072916204 0.036500346 0.002975002 -0.170397366 -0.035269084
## [25,] -0.256720184 -0.156530604 0.169086617 -0.138046792 -0.048895259
## [26,] 0.051918964 0.296741409 -0.005156454 -0.083009678 0.216571494
## [27,] 0.084359415 0.100688020 -0.114227791 -0.148738103 0.208138612
## [28,] 0.120369258 -0.077344549 0.058605865 -0.118307810 0.369882158
## [29,] -0.017476550 0.098581297 0.290640171 -0.123374300 0.136778131
## [30,] 0.071155368 0.358980614 -0.045365747 0.098313101 0.044415991
## [31,] 0.079238872 0.012357758 -0.006895099 0.168388181 0.232763624
## [32,] -0.028086257 0.128976176 -0.048914093 -0.244693795 0.065478758
## [33,] -0.313425032 0.017200560 0.025756825 0.056665381 -0.019064088
## [34,] 0.090324161 0.036390681 0.048763779 0.079261251 -0.182774450
## [35,] 0.194667953 0.343389070 -0.085604404 -0.057083611 0.058041118
## [36,] -0.013416299 0.053232165 -0.155293980 0.083629071 -0.035955754
## [37,] -0.072130518 -0.193744109 -0.012539500 -0.049838277 -0.049560311
## [38,] 0.044999381 0.011204272 -0.103756290 0.092586309 -0.287551306
## [39,] 0.061150489 0.024125475 -0.142951631 -0.229385150 -0.256943121
## [40,] -0.265377655 0.097062065 0.093014215 -0.213699531 -0.022636709
## [41,] 0.051439416 0.009663853 0.055125194 0.024795029 0.172635139
## [42,] -0.041059823 -0.134853410 0.172312932 0.039954393 0.160704027
## [43,] 0.067628298 -0.078310239 -0.096919879 -0.031034375 0.214601213
## [44,] -0.054041292 -0.115630361 -0.103497075 0.043803861 -0.030808591
## [45,] -0.156878132 0.157100915 0.094065543 0.172780733 0.036943326
## [46,] 0.028831197 -0.151257218 -0.195309950 0.016718701 0.010317848
## [47,] 0.044768598 -0.170529642 0.151432981 -0.035378367 0.155381862
## [48,] 0.096404055 0.038747503 0.230939224 0.159622208 -0.219161511
## [49,] 0.222727430 -0.010315626 -0.029405712 -0.091717767 -0.272315390
## [50,] 0.281934035 0.167270306 0.204337433 0.087660553 -0.033825316biplot(pca.x, col = c(2, 4))
Cols = function(vec){
cols = rainbow(length(unique(vec)))
+ return (cols[as.numeric(as.factor(vec))])
}
plot(pca.x$x[,1:2],col = Cols(k_col), pch =19, xlab ="Z1", ylab =" Z2")
#(c)
km.k3 <- kmeans(x,3,nstart = 100)
km.k3## K-means clustering with 3 clusters of sizes 20, 20, 20
##
## Cluster means:
## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## 1 1.309855 1.5297578 1.1779362 1.0127173 1.1392161 1.126534 1.2565549
## 2 2.475264 2.2168702 2.9349253 2.4655435 2.1325087 2.614003 1.9749243
## 3 0.649094 0.4272122 0.3176989 0.2322788 0.3968267 0.466004 0.6723788
## [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## 1 1.2520303 1.1322457 0.8168329 1.1008661 1.0163088 1.207543 1.1516678
## 2 2.5297753 2.2319161 2.7398656 2.2545316 2.6359340 2.547966 2.6918660
## 3 0.3036237 0.2740587 0.6096622 0.6082911 0.3533017 0.862027 0.3103367
## [,15] [,16] [,17] [,18] [,19] [,20] [,21]
## 1 1.1807631 1.0831205 1.0491036 1.3404711 0.95348547 0.1843175 1.2996476
## 2 2.3339909 2.4653947 2.1422365 2.7265256 2.07489790 2.8289318 2.4326758
## 3 0.5646508 0.6014428 0.6664156 0.5242812 -0.02815256 0.7511646 0.3753776
## [,22] [,23] [,24] [,25] [,26] [,27] [,28]
## 1 1.025534 1.4607360 0.9940398 1.0590802 1.1322356 1.0087072 1.1361883
## 2 2.388767 1.9959622 2.3270780 2.2484951 2.2870269 2.5202806 2.3146460
## 3 0.444185 0.5862425 0.4881267 0.2986866 0.4242547 0.6580425 0.5569209
## [,29] [,30] [,31] [,32] [,33] [,34] [,35]
## 1 1.7040022 1.3632733 0.8239648 1.320278 1.5653810 0.9339115 1.0138445
## 2 2.5695387 2.3448948 2.2700875 2.628201 1.9682686 2.5716311 2.2965644
## 3 0.1579515 0.7280993 1.0807349 1.055913 0.3004261 0.3584250 0.7454241
## [,36] [,37] [,38] [,39] [,40] [,41] [,42]
## 1 1.2694280 1.2306369 1.1967791 1.6208508 1.1248914 1.27757363 1.4278667
## 2 2.4356080 2.3580215 2.5349614 2.6495192 2.6114833 1.99571553 2.1598609
## 3 0.6046648 0.3827459 0.3997692 0.5030772 0.5984623 -0.07758204 0.2804351
## [,43] [,44] [,45] [,46] [,47] [,48] [,49]
## 1 0.9946385 1.1585822 1.3213006 1.0960575 1.4631512 1.1495011 1.2320374
## 2 2.2230252 2.4162878 2.6572722 2.1466762 2.0968924 2.2132917 2.2889500
## 3 0.6423291 0.7560256 0.3578973 0.8022603 0.5809075 0.6496547 0.2696592
## [,50]
## 1 1.2560154
## 2 2.0582427
## 3 0.7619741
##
## Clustering vector:
## [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3
## [36] 3 3 3 3 3 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1
##
## Within cluster sum of squares by cluster:
## [1] 877.2126 902.4400 935.7357
## (between_SS / total_SS = 41.5 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss"
## [5] "tot.withinss" "betweenss" "size" "iter"
## [9] "ifault"km.k3.table <- table(k,km.k3$cluster)
km.k3.table##
## k 1 2 3
## CLUSTER A 0 20 0
## CLUSTER B 1 0 19
## CLUSTER C 19 0 1#The row labels are the true cluster assignments for each observation and the columns labels are the kmean cluster asignments.
#In this format correct kmeans clustering assignment would show only one 20 value in any row or column. There for we can see
#that there are 2 errors in the kmeans clustering assignment.
#(d)
km.k2 <- kmeans(x,2,nstart = 100)
km.k2## K-means clustering with 2 clusters of sizes 40, 20
##
## Cluster means:
## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## 1 0.9794745 0.978485 0.7478175 0.6224981 0.7680214 0.7962691 0.9644669
## 2 2.4752637 2.216870 2.9349253 2.4655435 2.1325087 2.6140035 1.9749243
## [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## 1 0.777827 0.7031522 0.7132475 0.8545786 0.6848053 1.034785 0.7310023
## 2 2.529775 2.2319161 2.7398656 2.2545316 2.6359340 2.547966 2.6918660
## [,15] [,16] [,17] [,18] [,19] [,20] [,21]
## 1 0.872707 0.8422817 0.8577596 0.9323762 0.4626665 0.467741 0.8375126
## 2 2.333991 2.4653947 2.1422365 2.7265256 2.0748979 2.828932 2.4326758
## [,22] [,23] [,24] [,25] [,26] [,27] [,28]
## 1 0.7348595 1.023489 0.7410833 0.6788834 0.7782452 0.8333748 0.8465546
## 2 2.3887665 1.995962 2.3270780 2.2484951 2.2870269 2.5202806 2.3146460
## [,29] [,30] [,31] [,32] [,33] [,34] [,35]
## 1 0.9309769 1.045686 0.9523499 1.188096 0.9329035 0.6461682 0.8796343
## 2 2.5695387 2.344895 2.2700875 2.628201 1.9682686 2.5716311 2.2965644
## [,36] [,37] [,38] [,39] [,40] [,41] [,42]
## 1 0.9370464 0.8066914 0.7982742 1.061964 0.8616768 0.5999958 0.8541509
## 2 2.4356080 2.3580215 2.5349614 2.649519 2.6114833 1.9957155 2.1598609
## [,43] [,44] [,45] [,46] [,47] [,48] [,49]
## 1 0.8184838 0.9573039 0.8395989 0.9491589 1.022029 0.8995779 0.7508483
## 2 2.2230252 2.4162878 2.6572722 2.1466762 2.096892 2.2132917 2.2889500
## [,50]
## 1 1.008995
## 2 2.058243
##
## Clustering vector:
## [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##
## Within cluster sum of squares by cluster:
## [1] 2101.808 902.440
## (between_SS / total_SS = 35.3 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss"
## [5] "tot.withinss" "betweenss" "size" "iter"
## [9] "ifault"km.k2.table <- table(k,km.k2$cluster)
km.k2.table##
## k 1 2
## CLUSTER A 0 20
## CLUSTER B 20 0
## CLUSTER C 20 0#This time the kmeans clustering function has completely clustered together 2 of the 3 original clusters that have the closest
#mean speration [1.2-0.5=0.7].
#(e)
km.k4 <- kmeans(x,4,nstart = 100)
km.k4## K-means clustering with 4 clusters of sizes 11, 19, 20, 10
##
## Cluster means:
## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## 1 0.9684234 0.3104380 0.2573813 0.8701532 0.9640425 0.08611265 0.6398039
## 2 1.3390477 1.5762877 1.2286939 1.0778906 1.1629171 1.13801962 1.2230211
## 3 2.4752637 2.2168702 2.9349253 2.4655435 2.1325087 2.61400346 1.9749243
## 4 0.3084418 0.5775117 0.3736324 -0.5151685 -0.1979035 0.92811533 0.8303432
## [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## 1 0.77248789 0.3738488 0.2310056 0.6569641 0.4477177 0.8078905 0.4290859
## 2 1.17057978 1.1047955 0.9317316 1.1470023 1.0751547 1.2454097 1.1125388
## 3 2.52977527 2.2319161 2.7398656 2.2545316 2.6359340 2.5479659 2.6918660
## 4 0.03746968 0.3022635 0.8285940 0.5163493 0.2039378 0.8841818 0.3381909
## [,15] [,16] [,17] [,18] [,19] [,20] [,21]
## 1 0.6605928 0.2689090 0.4100315 0.8345475 -0.1321308 0.8841748 0.3051126
## 2 1.1961687 1.1182988 1.0402508 1.3524986 0.9878381 0.1707013 1.3106438
## 3 2.3339909 2.4653947 2.1422365 2.7265256 2.0748979 2.8289318 2.4326758
## 4 0.4914554 0.9485591 1.0035271 0.2417551 0.1191174 0.5740394 0.5242033
## [,22] [,23] [,24] [,25] [,26] [,27] [,28]
## 1 0.3068591 0.9625196 0.3111984 0.1862789 -0.4517381 1.17819360 0.6572198
## 2 1.0585594 1.5253206 1.0313083 1.0992041 1.2273931 1.04686810 1.1128662
## 3 2.3887665 1.9959622 2.3270780 2.2484951 2.2870269 2.52028064 2.3146460
## 4 0.5906302 0.1370763 0.6625290 0.4221391 1.2778458 0.04843702 0.5488309
## [,29] [,30] [,31] [,32] [,33] [,34] [,35]
## 1 0.6278806 0.8743944 1.0587433 0.8341859 0.5258548 0.2737157 0.6620560
## 2 1.7859342 1.3869036 0.8067553 1.3504132 1.6141058 0.9800260 1.0421817
## 3 2.5695387 2.3448948 2.2700875 2.6282012 1.9682686 2.5716311 2.2965644
## 4 -0.3600361 0.5857944 1.1119467 1.2689925 0.0863730 0.4215364 0.8101305
## [,36] [,37] [,38] [,39] [,40] [,41] [,42]
## 1 0.9099156 0.4114976 0.85664455 0.6038263 0.815035 0.5345661 0.0801116
## 2 1.1978760 1.2409721 1.18988905 1.5986688 1.126306 1.2404264 1.3640089
## 3 2.4356080 2.3580215 2.53496138 2.6495192 2.611483 1.9957155 2.1598609
## 4 0.4713142 0.4162712 -0.01000155 0.5461764 0.410188 -0.5448498 0.7368640
## [,43] [,44] [,45] [,46] [,47] [,48]
## 1 0.6388868 0.7168863 0.83765127 1.109522 1.186103308 0.5836007
## 2 0.9769967 1.2057861 1.33073984 1.096371 1.463939417 1.2272089
## 3 2.2230252 2.4162878 2.65727223 2.146676 2.096892407 2.2132917
## 4 0.7148658 0.7496473 -0.09142632 0.493057 0.001919018 0.6246537
## [,49] [,50]
## 1 0.08989182 1.2203810
## 2 1.33194302 1.1780466
## 3 2.28895004 2.0582427
## 4 0.37382039 0.4552712
##
## Clustering vector:
## [1] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 4 4 1 1 1 4 4 1 1 4 1 4 4 1
## [36] 4 1 4 1 4 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2
##
## Within cluster sum of squares by cluster:
## [1] 463.2582 834.2581 902.4400 414.4985
## (between_SS / total_SS = 43.7 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss"
## [5] "tot.withinss" "betweenss" "size" "iter"
## [9] "ifault"km.k4.table <- table(k,km.k4$cluster)
km.k4.table##
## k 1 2 3 4
## CLUSTER A 0 0 20 0
## CLUSTER B 10 0 0 10
## CLUSTER C 1 19 0 0#This time the kmeans clustering function has divided cluster B into two new clusters.
#(f)
km.pca.k3 <- kmeans(pca.x$x[,1:2],3,nstart = 100)
km.pca.k3## K-means clustering with 3 clusters of sizes 21, 20, 19
##
## Cluster means:
## PC1 PC2
## 1 -1.410705 -0.8315563
## 2 7.366597 0.3233194
## 3 -6.195112 0.5787523
##
## Clustering vector:
## [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3
## [36] 3 3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##
## Within cluster sum of squares by cluster:
## [1] 73.25643 86.38012 55.48136
## (between_SS / total_SS = 89.7 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss"
## [5] "tot.withinss" "betweenss" "size" "iter"
## [9] "ifault"km.pca.k3.table <- table(k,km.pca.k3$cluster)
km.pca.k3.table##
## k 1 2 3
## CLUSTER A 0 20 0
## CLUSTER B 1 0 19
## CLUSTER C 20 0 0#This kmeans cluster analysis using princple components 1 and 2 only shows similar accuracy to the kmeans cluster analysis
#in a which we would expect given that all variables have the same random spread and the same mean shift.
#(g)
x.scaled <- scale(x, scale=T)
km.k3.scaled <- kmeans(x.scaled,3,nstart = 100)
km.k3.scaled## K-means clustering with 3 clusters of sizes 21, 19, 20
##
## Cluster means:
## [,1] [,2] [,3] [,4] [,5] [,6]
## 1 -0.1039736 0.0883601 -0.2754701 -0.1087234 -0.07246093 -0.2001155
## 2 -0.6993348 -0.7893349 -0.7328472 -0.7891656 -0.72699079 -0.7416277
## 3 0.7735403 0.6570901 0.9854484 0.8638669 0.76672522 0.9146676
## [,7] [,8] [,9] [,10] [,11] [,12]
## 1 -0.05695131 -0.06555976 -0.04070619 -0.4138295 -0.2156899 -0.2315845
## 2 -0.60788895 -0.77346682 -0.94387071 -0.5607947 -0.5513707 -0.7491591
## 3 0.63729337 0.80363122 0.93941867 0.9672759 0.7502766 0.9548648
## [,13] [,14] [,15] [,16] [,17] [,18]
## 1 -0.3394769 -0.1734508 -0.1033403 -0.2716986 -0.2068716 -0.1536902
## 2 -0.5618599 -0.8065449 -0.6978473 -0.6805532 -0.5788269 -0.7484202
## 3 0.8902177 0.9483410 0.7714623 0.9318091 0.7671007 0.8723739
## [,19] [,20] [,21] [,22] [,23] [,24]
## 1 -0.05056514 -0.6915034 -0.1343497 -0.2202581 0.1314797 -0.1842246
## 2 -0.81174696 -0.3803478 -0.7504692 -0.6819561 -0.7421851 -0.6597100
## 3 0.82425301 1.0874090 0.8540129 0.8791293 0.5670222 0.8201604
## [,25] [,26] [,27] [,28] [,29] [,30]
## 1 -0.1457785 -0.2204352 -0.2549129 -0.1429850 0.1016353 -0.02904591
## 2 -0.7098896 -0.5441009 -0.6354125 -0.7908161 -0.9830227 -0.76921815
## 3 0.8274626 0.7483528 0.8713004 0.9014095 0.8271545 0.76125545
## [,31] [,32] [,33] [,34] [,35] [,36]
## 1 -0.4780791 -0.2823916 0.2086279 -0.2456009 -0.3201466 -0.1740479
## 2 -0.2438967 -0.4444246 -0.8413057 -0.7371824 -0.4747019 -0.5893891
## 3 0.7336849 0.7187146 0.5801811 0.9582042 0.7871207 0.7426700
## [,37] [,38] [,39] [,40] [,41] [,42]
## 1 -0.09152871 -0.1930792 -0.01455523 -0.2337352 0.1489961 -0.0478431
## 2 -0.77311624 -0.7224607 -0.87726224 -0.7382061 -0.9377189 -0.6856784
## 3 0.83056558 0.8890708 0.84868212 0.9467178 0.7343870 0.7016297
## [,43] [,44] [,45] [,46] [,47] [,48]
## 1 -0.2785215 -0.2458261 -0.1029970 -0.1844529 0.05346993 -0.1668043
## 2 -0.5965281 -0.6176396 -0.8006893 -0.5417193 -0.73601746 -0.6928590
## 3 0.8591492 0.8448750 0.8688016 0.7083089 0.64307316 0.8333605
## [,49] [,50]
## 1 -0.06632607 -0.1318751
## 2 -0.75751895 -0.5970250
## 3 0.78928537 0.7056426
##
## Clustering vector:
## [1] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2
## [36] 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##
## Within cluster sum of squares by cluster:
## [1] 609.3490 560.5724 576.5949
## (between_SS / total_SS = 40.8 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss"
## [5] "tot.withinss" "betweenss" "size" "iter"
## [9] "ifault"km.k3.scaled.table <- table(k,km.k3.scaled$cluster)
km.k3.scaled.table##
## k 1 2 3
## CLUSTER A 0 0 20
## CLUSTER B 1 19 0
## CLUSTER C 20 0 0#This kmeans cluster analysis using princple components 1 and 2 only shows similar accuracy to the kmeans cluster analysis
#in a which we would expect given that all variables have the same random spread and the same mean shift.